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## Fibonacci Numbers

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... They show up in nature, they show up in math, and they've got some beautiful properties.

# Level 3

Compute $\frac{1}{2} + \frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \dots + \frac{F_k}{2^k} + \dots$ where $$F_k$$ represents the the $$k^\text{th}$$ term of the Fibonacci sequence: $$1, 1, 2, 3, 5, 8, 13, ...$$

Find the last digit of the 123456789-th Fibonacci number.

In the Fibonacci sequence, $$F_{0}=1$$, $${F_1}=1$$, and for all $$N>1$$, $$F_N=F_{N-1}+F_{N-2}$$.

How many of the first 2014 Fibonacci terms end in 0?

The Fibonacci sequence is defined by $$F_1 = 1, F_2 = 1$$ and $$F_{n+2} = F_{n+1} + F_{n}$$ for $$n \geq 1$$.

Find the greatest common divisor of $$F_{484}$$ and $$F_{2013}$$.

$\sum_{n=0}^{\infty} \frac{F_{n}}{3^{n}}= \ ?$

Details and Assumptions

$$F_{n}$$ is the $$n^\text{th}$$ Fibonacci number, with $$F_{1} = F_{2} = 1$$.

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